Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 ↓


Trigonometry


Trigonometry is a branch of mathematics that studies the relationships between the side lengths and angles of triangles. It is incredibly useful in many fields, including engineering, physics, architecture, and even game development. Let's explore the fascinating world of trigonometry in simple terms, enhanced with visual representations and examples!

Basic concepts

The word "trigonometry" comes from two Greek words, "trigonon" (which means triangle) and "metron" (which means measurement). Therefore, trigonometry is essentially the study of triangles. We mainly focus on right triangles, which have an angle of 90 degrees.

Parts of a right triangle

A right-angled triangle has three sides:

  • Hypotenuse: The longest side in a triangle opposite the right angle.
  • Opposite: The direction opposite to the angle you are focusing on (other than a right angle).
  • Adjacent: The side you're focusing on (not the hypotenuse).

Trigonometric ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. They are fundamental in solving many geometric problems.

Sine, cosine and tangent

The three primary trigonometric ratios are sine, cosine, and tangent. Here are their definitions based on a right triangle:

Sine of an angle = opposite / hypotenuse
Cosine of an angle (cos) = adjacent / hypotenuse
Tangent of an angle (tan) = opposite / adjacent

Let's look at this with an example: a triangle ABC in which angle C is 90 degrees. If angle A is the angle we are interested in, then:

sin(A) = opposite side of A / hypotenuse
cos(A) = adjacent side/hypotenuse of A
tan(A) = opposite side of A / adjacent side of A

Here's a visual of what a right triangle would look like:



  
  C
  B
  A
  
  
  
  near
  Ear
  Antonym

Using trigonometric ratios

Trigonometric ratios can be used to find missing side lengths and angles in right triangles. Let's look at some examples.

Example 1: Finding a side

Suppose we have a right triangle whose angle A is 30 degrees and the hypotenuse is 10 units. We want to find the side opposite to angle A.

Using sin(A) = opposite / hypotenuse,
sin(30) = opposite / 10
0.5 = opposite / 10
Opposite = 0.5 * 10 = 5

Thus, the side opposite angle A is 5 units long.

Example 2: Finding the angle

Consider another triangle, in which the adjacent side is 4 units, and the opposite side is 3 units. We want to find angle A using the tangent ratio.

Using tan(A) = opposite/adjacent,
tan(a) = 3 / 4
a = arctan(3/4)

Using a calculator, you can figure out that angle A is approximately 36.87 degrees.

Special angles

Trigonometry often refers to special angles, typically 0, 30, 45, 60, and 90 degrees. These angles have known sine, cosine, and tangent values.

Angle (degrees) | sin | cos | tan
,
0 | 0 | 1 | 0
30 | 0.5 | √3/2 | 1/√3
45 | √2/2 | √2/2 | 1
60 | √3/2 | 0.5 | √3
90 | 1 | 0 | Undetermined

It is often beneficial to memorize these values in order to quickly solve trigonometric problems.

Pythagorean identity

In trigonometry, an important identity arises from the Pythagorean theorem. For any angle θ, the following is always true:

(sin(θ))^2 + (cos(θ))^2 = 1

This is known as the Pythagorean identity and can be derived from the Pythagorean theorem applied to the unit circle.

Solving real-life problems

Trigonometry isn't just theoretical; it helps solve real-world problems. Here are some examples:

Example 1: Measuring height

Suppose you want to determine the height of a tree. You stand 10 m away from it and measure the angle of elevation (angle from the ground) from the top of the tree as 30 degrees. We can use the tangent ratio to find the height.

Using tan(θ) = opposite/adjacent,
tan(30) = height / 10
Height = tan(30) * 10 = (√3/3) * 10 ≈ 5.77

The height of the tree is about 5.77 meters.

Example 2: Navigation

In navigation, trigonometry helps determine position and course. For example, if a ship travels 100 km south and then 75 km east, the use of the Pythagorean theorem allows you to calculate the direct distance back to the starting point.

Let a = 100 km (south) and b = 75 km (east).
d^2 = a^2 + b^2
d^2 = 100^2 + 75^2 
d^2 = 10000 + 5625 
D = √15625
d = 125 km

The direct distance is 125 km.

Conclusion

Understanding trigonometry is the key to solving many practical and theoretical problems. By familiarizing yourself with the basic ratios, identities, and applications, you will be better equipped to tackle the challenges of mathematics and beyond.

Whether you're calculating angles, finding side lengths, or working with real-world scenarios, the principles you learn in trigonometry will be a valuable tool in your problem-solving toolkit!


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